Torsion of rational elliptic curves over cubic fields and sporadic points on $X_1(n)$

We classify the possible torsion structures of rational elliptic curves over cubic fields. Along the way we find a previously unknown torsion structure over a cubic field, $\mathbb{Z}\mathrm{/}21\mathbb{Z}$, which corresponds to a sporadic point on $X_1(21)$ of degree 3, which is the lowest possible degree of a sporadic point on a modular curve $X_1(n)$.